| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2014 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find value where max/min occurs |
| Difficulty | Standard +0.3 This is a standard harmonic form question requiring routine application of R cos(θ + α) conversion, solving a trigonometric equation, and finding maximum values. Part (iii) involves a simple substitution and recognizing that the maximum occurs when cos equals 1. All techniques are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(R = 13\) | B1 | |
| Use appropriate formula to find \(\alpha\) | M1 | |
| Obtain 67.38° | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to find at least one value of \(\cos^{-1}\frac{8}{13}\) or \(\cos^{-1}\frac{8}{R}\) | M1 | |
| Obtain one correct value of \(\theta\) (240.6 or 344.6) | A1 | |
| Carry out correct method to find second value of \(\theta\) within the range | DM1 | |
| Obtain second correct value (344.6 or 240.6) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| State or imply \(7 + 13\cos(\frac{1}{2}\phi + 67.38)\) following their answers from part (i) | B1 | |
| State 20 | B1 | |
| Attempt to find \(\phi\) for which \(\cos(\frac{1}{2}\phi + 67.38) = 1\) | M1 | |
| Obtain 585.2 | A1 | [4] |
**(i)**
State or imply $R = 13$ | B1 |
Use appropriate formula to find $\alpha$ | M1 |
Obtain 67.38° | A1 | [3]
**(ii)**
Attempt to find at least one value of $\cos^{-1}\frac{8}{13}$ or $\cos^{-1}\frac{8}{R}$ | M1 |
Obtain one correct value of $\theta$ (240.6 or 344.6) | A1 |
Carry out correct method to find second value of $\theta$ within the range | DM1 |
Obtain second correct value (344.6 or 240.6) | A1 | [4]
**(iii)**
State or imply $7 + 13\cos(\frac{1}{2}\phi + 67.38)$ following their answers from part (i) | B1 |
State 20 | B1 |
Attempt to find $\phi$ for which $\cos(\frac{1}{2}\phi + 67.38) = 1$ | M1 |
Obtain 585.2 | A1 | [4]7 (i) Express $5 \cos \theta - 12 \sin \theta$ in the form $R \cos ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the value of $\alpha$ correct to 2 decimal places.\\
(ii) Hence solve the equation $5 \cos \theta - 12 \sin \theta = 8$ for $0 ^ { \circ } < \theta < 360 ^ { \circ }$.\\
(iii) Find the greatest possible value of
$$7 + 5 \cos \frac { 1 } { 2 } \phi - 12 \sin \frac { 1 } { 2 } \phi$$
as $\phi$ varies, and determine the smallest positive value of $\phi$ for which this greatest value occurs.\\[0pt]
[4]
\hfill \mbox{\textit{CAIE P2 2014 Q7 [11]}}